Algebra Examples

$\frac{4 + \sqrt{6}}{\sqrt{2} + \sqrt{3}}$

Step 1

Multiply $\frac{4 + \sqrt{6}}{\sqrt{2} + \sqrt{3}}$ by $\frac{\sqrt{2} - \sqrt{3}}{\sqrt{2} - \sqrt{3}}$ .

$\frac{4 + \sqrt{6}}{\sqrt{2} + \sqrt{3}} \cdot \frac{\sqrt{2} - \sqrt{3}}{\sqrt{2} - \sqrt{3}}$

Step 2

Combine fractions.

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Step 2.1

Multiply $\frac{4 + \sqrt{6}}{\sqrt{2} + \sqrt{3}}$ by $\frac{\sqrt{2} - \sqrt{3}}{\sqrt{2} - \sqrt{3}}$ .

$\frac{(4 + \sqrt{6}) (\sqrt{2} - \sqrt{3})}{(\sqrt{2} + \sqrt{3}) (\sqrt{2} - \sqrt{3})}$

Step 2.2

Expand the denominator using the FOIL method.

$\frac{(4 + \sqrt{6}) (\sqrt{2} - \sqrt{3})}{{\sqrt{2}}^{2} - \sqrt{6} + \sqrt{6} - {\sqrt{3}}^{2}}$

Step 2.3

Simplify.

$\frac{(4 + \sqrt{6}) (\sqrt{2} - \sqrt{3})}{- 1}$

Step 2.4

Simplify the expression.

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Step 2.4.1

Move the negative one from the denominator of $\frac{(4 + \sqrt{6}) (\sqrt{2} - \sqrt{3})}{- 1}$ .

$- 1 \cdot ((4 + \sqrt{6}) (\sqrt{2} - \sqrt{3}))$

Step 2.4.2

Rewrite $- 1 \cdot ((4 + \sqrt{6}) (\sqrt{2} - \sqrt{3}))$ as $- ((4 + \sqrt{6}) (\sqrt{2} - \sqrt{3}))$ .

$- ((4 + \sqrt{6}) (\sqrt{2} - \sqrt{3}))$

Step 3

Expand $(4 + \sqrt{6}) (\sqrt{2} - \sqrt{3})$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$- (4 (\sqrt{2} - \sqrt{3}) + \sqrt{6} (\sqrt{2} - \sqrt{3}))$

Step 3.2

Apply the distributive property.

$- (4 \sqrt{2} + 4 (- \sqrt{3}) + \sqrt{6} (\sqrt{2} - \sqrt{3}))$

Step 3.3

Apply the distributive property.

$- (4 \sqrt{2} + 4 (- \sqrt{3}) + \sqrt{6} \sqrt{2} + \sqrt{6} (- \sqrt{3}))$

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $- 1$ by $4$ .

$- (4 \sqrt{2} - 4 \sqrt{3} + \sqrt{6} \sqrt{2} + \sqrt{6} (- \sqrt{3}))$

Step 4.1.2

Combine using the product rule for radicals.

$- (4 \sqrt{2} - 4 \sqrt{3} + \sqrt{6 \cdot 2} + \sqrt{6} (- \sqrt{3}))$

Step 4.1.3

Multiply $6$ by $2$ .

$- (4 \sqrt{2} - 4 \sqrt{3} + \sqrt{12} + \sqrt{6} (- \sqrt{3}))$

Step 4.1.4

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 4.1.4.1

Factor $4$ out of $12$ .

$- (4 \sqrt{2} - 4 \sqrt{3} + \sqrt{4 (3)} + \sqrt{6} (- \sqrt{3}))$

Step 4.1.4.2

Rewrite $4$ as $2^{2}$ .

$- (4 \sqrt{2} - 4 \sqrt{3} + \sqrt{2^{2} \cdot 3} + \sqrt{6} (- \sqrt{3}))$

Step 4.1.5

Pull terms out from under the radical.

$- (4 \sqrt{2} - 4 \sqrt{3} + 2 \sqrt{3} + \sqrt{6} (- \sqrt{3}))$

Step 4.1.6

Multiply $\sqrt{6} (- \sqrt{3})$ .

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Step 4.1.6.1

Combine using the product rule for radicals.

$- (4 \sqrt{2} - 4 \sqrt{3} + 2 \sqrt{3} - \sqrt{6 \cdot 3})$

Step 4.1.6.2

Multiply $6$ by $3$ .

$- (4 \sqrt{2} - 4 \sqrt{3} + 2 \sqrt{3} - \sqrt{18})$

Step 4.1.7

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 4.1.7.1

Factor $9$ out of $18$ .

$- (4 \sqrt{2} - 4 \sqrt{3} + 2 \sqrt{3} - \sqrt{9 (2)})$

Step 4.1.7.2

Rewrite $9$ as $3^{2}$ .

$- (4 \sqrt{2} - 4 \sqrt{3} + 2 \sqrt{3} - \sqrt{3^{2} \cdot 2})$

Step 4.1.8

Pull terms out from under the radical.

$- (4 \sqrt{2} - 4 \sqrt{3} + 2 \sqrt{3} - (3 \sqrt{2}))$

Step 4.1.9

Multiply $3$ by $- 1$ .

$- (4 \sqrt{2} - 4 \sqrt{3} + 2 \sqrt{3} - 3 \sqrt{2})$

Step 4.2

Subtract $3 \sqrt{2}$ from $4 \sqrt{2}$ .

$- (\sqrt{2} - 4 \sqrt{3} + 2 \sqrt{3})$

Step 4.3

Add $- 4 \sqrt{3}$ and $2 \sqrt{3}$ .

$- (\sqrt{2} - 2 \sqrt{3})$

Step 5

Apply the distributive property.

$- \sqrt{2} - (- 2 \sqrt{3})$

Step 6

Multiply $- 2$ by $- 1$ .

$- \sqrt{2} + 2 \sqrt{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- \sqrt{2} + 2 \sqrt{3}$

Decimal Form:

$2.04988805 \dots$